1. Discrete Dependent Variables Linear Regression, Dummy Variables If discrete dependent variable: need new technique Examples: Firm join Energy Star or not. Parcel of land developed as urban, agriculture, or open space. Species goes extinct or not. We’ll focus on: “Binary Choice Estimators”

2. Example: Tumors and ETU Big question: How does exposure affect chance of contracting disease? Treated foods contain ETU – may be harmful to health. Some rats exposed to ETU contracted tumors. 1. How does prob of tumor depend on dose? 2. What dose associated with 10% tumor rate (To advise on regulation)?

3. Evidence 6 dose groups (0,5,25,125,250,500) ~70 rats per group.

4. How ‘bout a Linear Model? Linear Model: Y=+X+

5. Problems with Linear Model How do we interpret dependent variable? (“chance of tumor?”) If Dose=0, chance of tumor < 0. If Dose large, chance of tumor > 1. Doesn’t make sense, and chance is linear in dose.

6. Binary Choice Models Logit (based on logistic cdf) and Probit (based on Normal cdf). Logistic cdf: Draw on board.

7. Adding Explanatory Variables Interpretation of Binary Choice Estimator: Probability of “Yes”. Replace “X” with function of explanatory variables:

8. Probability of Tumor

9. Dose @ 10% Tumor Chance What dose gives a 10% chance of contracting a tumor? After a bunch of math (see handout), D=170.24